Customer Review

November 27, 2013
This inconsequential collection of articles tries to pass itself off as much more with vastly overblown pretensions and sensationalism.

Chemla's introduction speaks for example of "the fundamental questions raised" by the editorial practices of Heiberg in his authoritative editions of classical Greek works, such as: "how far is his vision of Euclidean proof, formed at the end of the nineteenth century, conveyed through the text of his critical edition?" (p. 21) Quite far indeed it might seem from statements such as "Netz reveals [that] Heiberg himself decided to give some propositions the status of postulate and others that of definition, with the manuscript containing nothing of the sort" (p. 26).

Let us trace what this dramatic and grandiose statement ultimately amounts to. One has to read Netz's chapter quite carefully not to miss the two or three sentences on which Chemla's exclamation is based; the basis for it is mentioned in passing on p. 194, where Netz laments that Heiberg marked the headings "definitions" and "postulates" by "om. for 'the manuscripts omitted...'" rather than "add. for 'I added'".

If we want to know the statements of the definitions and postulates in question then we have to turn to Archimedes's text itself. In Netz's edition of Archimedes's works the relevant passages occur in vol. 1, pp. 34-36. We see there that the "propositions" in question occur in Archimedes's introduction prefaced by the phrase "First are written the principles and assumptions required for the proofs of these properties." Then follows the supposedly intrusive and prejudiced heading of "definitions." All statements under this heading are very obviously intended as definitions. They say things like "I call concave in the same direction such a line..." Having listed his definitions Archimedes writes "And I assume these:" Again this is followed by the supposedly anachronistic and biased heading of "Postulates", followed by postulates such as "That among lines which have the same limits, the straight line is the smallest."

Clearly, then, it is ridiculous to characterise these statements as "propositions", which only "Heiberg himself" has "decided to give the status of definitions and postulates". Thus we see that "the amazing Prologue by the editor, Karine Chemla" (as Hilary Putnam calls it in a blurb printed before the title page) is engaged in quite shameless sensationalism.

A similar instance is Chemla's claim that "Heiberg removed any hint of the actors' ways of drawing and using figures" (p. 24). This is certainly a wild exaggeration, and it is in any case erroneous to attribute this practice to Heiberg since his figures are the commonsensical ones used in all printed editions of Euclid since hundreds of years before Heiberg was even born (cf. p. 136).

It is true that Greek manuscript figures are very schematic and crude, and often compromise generality and representative accuracy for ease of drawing. This was most likely due to limitations of their drawing skills, tools and media, but such a sensible and commonsensical interpretation is not going to generate fancy revisionist publications and get you a professorship at Stanford, so Netz takes another view:

"A major claim of my book [The Shaping of Deduction in Greek Mathematics] was that diagrams play a role in Greek mathematical reasoning. I have suggested there---following Poincare---that the diagrams may have been used as if they were merely topological. My consequent study of the palaeography of Greek diagrams has revealed a striking and more powerful result: the diagrams, at least as preserved by early Byzantine manuscripts, simply were topological. Heiberg's choice to obscure this character of the diagrams was not only philologically but also philosophically motivated. Clearly, he did not perceive diagrams to form part of the logic of the text and for that reason, on the one hand, he did not value them enough to care for their proper edition and, on the other hand, preferred to produce them as mere 'illustrations'---as visual aids revealing to the mind a picture of the object under discussion. The implication---false for Archimedes as for Greek mathematics more generally---would be that the text is logically self-enclosed, that all claims are textually explicit." (pp. 175-176)

You can decide for yourself whether to laugh or cry at these wild extrapolations, but you may be interested to know, what Netz does not tell you, that this is in fact not a strengthening of his previous position but rather an explicit contradiction of it. In the very book he himself refers to here, Netz wrote that the Greek mathematical diagram "contains at least one other type of information [beyond the topological], namely the straightness of lines; that points stand 'on a line' is constantly assumed on the basis of the diagram ... The man-made diagram, unlike nature's shapes, was governed by the distinction between straight and non-straight alone" (Shaping of Deduction, p. 34). The fact that Netz does not bother to discuss his previous opinion even when he is explicitly contradicting it is perhaps a reflection of how poorly supported any of his pretentious speculations ever are. (Indeed, in another chapter we see that "we also have at least one example of straight lines being used to represent a curved line", namely "a parabola is represented by an isosceles triangle" in the Archimedes Palimpsest; pp. 147-148.)
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